Dear Aderemi,

I believe a positive answer to your question can be derived for Azumaya algebras using the results of Dwyer and Friedlander in their paper:  Étale K-theory of Azumaya algebras.  Proceedings of the Luminy conference on algebraic K-theory (Luminy, 1983).  J. Pure Appl. Algebra 34 (1984), no. 2-3, 179-191.  

Dear Paul Arne,

Thanks for your comments.

However, the K-theory context for my question is Higher Algebraic K-theory (Quillen type) and not  Etale K-theory. Indeed the motivation for my question is the desire to have a non-commutative analogue of Soule's  etale Chern characters from Higher Algebraic K-theory of  the ring R of integers in a number field F to etale cohomology of R. (See  C. Soule: K-theorie des annex d'entiers de corps de nombers et cohomologie etale. Invent Math. 55 (1979) 252-295). Soule's construction and associated Chern characters are also discussed  in my article A. Kuku: "Higher Algebraic K-theory" Handbook of Algebra, Vol 4, 2006 Elsevier, 3-74, sections 5.3 and 6.2.

My ultimate non-commutative generalization of R is a non-commutative R-order A (e.g the groupring A=RG, G finite, non-commutative)  in a semi-simple F-algebra B (e.g; B= FG) and the goal is to construct an etale chern character from K_n(A) (which I understand very well from my previous work ) to a suitably defined etale cohomology of A.(which is so far elusive). As such, my goal would not be achieved through Dwyer-Friedlander construction of etale K-theory of Azumaya algebras..

Dear Aderemi,

A group ring RG is an Azumaya algebra if R is commutative and G has finite order invertible in R by a generalization of Maschke's theorem (this condition is not necessary however).  In particular, this applies to group rings of finite non-abelian groups over (some) rings of integers in number fields.

The following applies to Azumaya algebras:  Algebraic K-theory maps naturally to etale K-theory and hence to etale cohomology via the edge homomorphism in the Atiyah-Hirzebruch spectral sequence recorded as Proposition 1.6 in the mentioned paper of Dwyer and Friedlander (assumptions and details omitted). 

Dmitry Kaledin has worked on cohomology theories in the non-commutative setting: imperium.lenin.ru/~kaledin/math/kaledin-5ecm.pdf

On page three it is stated that there is no known extension of etale cohomology to the non-commutative setting. 

Dear Paul,

Thanks for your further comments and information which shows that such a construction is possible for group rings that are Azumaya algebras when one passes through etale K-theory. Actually,  one such group ring is RG where R is the ring of integers in a p-adic field F such that  FG splits and p is prime to the order of G. This  is a  highly restrictive  type of group ring. It is  also a maximal order in FG--and maximal orders are very useful for computations involving arbitrary orders (See my book: A. Kuku  "Representation Theory and Higher Algebraic K-theory, Chapman and Hall 2007; 7.1., 7.2.) However, one would like to be able to deal with more general situation involving arbirary orders and  maximal orders as follows--- provided  one can have an etale K-theory and etale cohomology for maximal orders C.

Let F be a number field or p-adic field with ring of integers R, A any R-order in a semi-simple F-algebra. Let C be a maximal order containing A. The inclusion A---> C induces a map K_n(A) ----> K_n(C) and one could possibly define maps map K_n(C)-----> etale K-theory of C------> etale cohomology of C restricted to A.  This construction (if possible ) is bound to be very cumbersome even though C behaves nicely as a product of matrix algebras over maximal orders in some division algebras.

However it seems to me more desirable to go directly from Higher algebraic K-theory of A to etale cohomology of A (like Soule has done with R)  and this  I have been trying  unsuccessfully to figure out how to do.

Dear Aderemi,

Thank you for the discussion.  Please keep us posted.

Dear Aderemi,

As we talked about a while ago, but forgot to record in this thread, blueprints for extending invariants from schemes to non-commutative settings have been worked out by Blanc

Article Topological K-theory of complex noncommutative Spaces

and by Carchedi & Elmanto

Preprint Relative \'{E}tale Realizations of Motivic Spaces and Dwyer-...

Best wishes,

Paul Arne

Dear Paul Arne,

Thank you very much for the valuable information. I have requested both papers from Blanc, Carchedi and Elmanto and will look at them very closely when I receive them.

Thanks once again,

Best wishes,

'Remi.